Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces
We introduce the setting of extended metric-topological
measure spaces as a general ``Wiener like'' framework for optimal
transport problems and nonsmooth metric analysis in
After a brief review of optimal transport tools for general Radon
we discuss the notions of the Cheeger energy,
of the Radon measures concentrated on absolutely continuous curves,
and of the induced ``dynamic transport
distances''. We study their main properties and their
links with the theory of Dirichlet forms and the
Bakry-\'Emery curvature condition,
in particular concerning the contractivity properties
and the EVI formulation of the induced Heat semigroup.
The paper is available on the
cvgmt preprint server.